15 research outputs found
Non-Markovian diffusion equations and processes: analysis and simulations
In this paper we introduce and analyze a class of diffusion type equations
related to certain non-Markovian stochastic processes. We start from the
forward drift equation which is made non-local in time by the introduction of a
suitable chosen memory kernel K(t). The resulting non-Markovian equation can be
interpreted in a natural way as the evolution equation of the marginal density
function of a random time process l(t). We then consider the subordinated
process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding
time evolution of the marginal density function of Y(t) is governed by a
non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We
develop several applications and derive the exact solutions. We consider
different stochastic models for the given equations providing path simulations.Comment: 43 pages, 19 figures, in press on Physica A (2008
Generation-by-Generation Dissection of the Response Function in Long Memory Epidemic Processes
In a number of natural and social systems, the response to an exogenous shock
relaxes back to the average level according to a long-memory kernel with . In the presence of an epidemic-like
process of triggered shocks developing in a cascade of generations at or close
to criticality, this "bare" kernel is renormalized into an even slower decaying
response function . Surprisingly, this means that the
shorter the memory of the bare kernel (the larger ), the longer the
memory of the response function (the smaller ). Here, we present a
detailed investigation of this paradoxical behavior based on a
generation-by-generation decomposition of the total response function, the use
of Laplace transforms and of "anomalous" scaling arguments. The paradox is
explained by the fact that the number of triggered generations grows
anomalously with time at so that the contributions of active
generations up to time more than compensate the shorter memory associated
with a larger exponent . This anomalous scaling results fundamentally
from the property that the expected waiting time is infinite for . The techniques developed here are also applied to the case
and we find in this case that the total renormalized response is a {\bf
constant} for followed by a cross-over to
for .Comment: 27 pages, 4 figure
Glycoproteins in Claudin-Low Breast Cancer Cell Lines Have a Unique Expression Profile
Claudin proteins are components of
epithelial tight junctions;
a subtype of breast cancer has been defined by the reduced expression
of mRNA for claudins and other genes. Here, we characterize the expression
of glycoproteins in breast cell lines for the claudin-low subtype
using liquid chromatography/tandem mass spectrometry. Unsupervised
clustering techniques reveal a group of claudin-low cell lines that
is distinct from nonmalignant, basal, and luminal lines. The claudin-low
cell lines express F11R, EPCAM, and other proteins at very low levels,
whereas CD44 is expressed at a high level. Comparison of mRNA expression
to glycoprotein expression shows modest correlation; the best agreement
occurs when the mRNA expression level is lowest and little or no protein
is detected. These findings from cell lines are compared to those
for tumor samples by the Clinical Proteomic Tumor Analysis Consortium
(CPTAC). The CPTAC samples contain a group low in CLDN3. The samples
low in CLDN3 proteins share many differentially expressed glycoproteins
with the claudin-low cell lines. In contrast to the situation for
cell lines or patient samples classified as claudin-low by RNA expression,
however, most of the tumor samples low in CLDN3 protein express the
estrogen receptor or HER2. These tumor samples express CD44 protein
at low rather than high levels. There is no correlation between CLDN3
gene expression and protein expression in these CPTAC samples; hence,
the claudin-low subtype defined by gene expression is not the same
group of tumors as that defined by low expression of CLDN3 protein
Inverse Stable Subordinators
The inverse stable subordinator provides a probability model for time-fractional differential equations, and leads to explicit solution formulae. This paper reviews properties of the inverse stable subordinator, and applications to a variety of problems in mathematics and physics. Several different governing equations for the inverse stable subordinator have been proposed in the literature. This paper also shows how these equations can be reconciled